[FOM] Space-Time (response to Timothy Chow's
Laura Elena Morales Guerrero
lem at fis.cinvestav.mx
Wed Feb 22 12:53:21 EST 2006
message Dated: Tue, 14 Feb 2006 19:12:26 -0500 (EST)
and sent To: fom at cs.nyu.edu)
Tim wrote: "As I think I've mentioned to Joe Shipman in private email
before, the part about his alternate history of physics and mathematics
that I have the hardest time believing is the notion that general
relativity would somehow make it more intuitively plausible that
translation invariance or rotational invariance of space is not true.
I recall that when I first studied general relativity, I was puzzled by
the equation T = 8*pi*G because of the following confused line of
thinking: I can take an arbitrary pseudomanifold, and compute its
curvature tensors, and then I guess Einstein's equation gives me the
stress-energy tensor, but "then what"? The equation appears to put no
constraints on what can happen.
With hindsight, I can diagnose my confusion as a case of thinking of
Einstein's equation as primarily a *mathematical* equation rather than as
a *physical* equation. Physically, mass/energy is conceptually distinct
from the curvature of spacetime. We might, for example, have some other
physical theory about certain kinds of energy (such as electromagnetic
energy) that can then be combined with general relativity to yield some
subtle physical predictions.
With regard to RVM, my point is that I cannot see how general relativity
would lead us to think that *space* is not isotropic. When space fails to
be isotropic, the physical thinking is that it's because something
(matter) is messing things up, not that space itself lacks symmetry.
I would be interested to hear from physicists as to what their intuition
is on this question."
[end of quote]
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Here are some thoughts in this respect.
Regards,
LE
COMPARISON
In Newtonian physics, we can speak of small objects moving in "space" over
"time". In spacetime, however, nothing physical "moves" at all! Rather,
we represent the entire history of motion of each small object by a curve
in spacetime, called the "world line". For example, in Newtonian physics,
we might say that a space capsule "moves in a circular orbit around the
earth". But in general theory of relativity (gtr) we would say that the
history of motion of the capsule corresponds to an (approximately helical)
curve in spacetime.
In Newtonian physics, the trajectory followed by the capsule (a circle, in
this example) is "curved" due to the "gravitational attraction" of a
massive object, the Earth. But in gtr, the spacetime near the Earth is
"curved" by the presence of this massive object, and the world line
representing the history of motion of a small object which is "freely
falling", i.e., feeling no forces, such as our space capsule, is a curve
which is "as straight as possible" in this curved spacetime. Such a curve
is called a "geodesic".
SPACETIME
Experiments show that all particles fall in the same way independently of
their charge, mass or any other property, we can conclude that all
possible trajectories form an independent structure. This structure is
what we call spacetime. Spacetime tells energy, matter and radiation how
to fall.
Space_Time (S-T) is not described by Minkowski S-T when gravity (matter)
is present. Gravity is curved S-T. Space is curved near masses. Space
behaves like a frictionless mattress (inside which we live) that pervades
everything. Massive objects pull the foam of the mattress towards them,
thus deforming the mattress shape. Energy also curves space. The curved
shape of spacetime is best described by the behaviour of the spacetime
distance (the metric) between two neighbouring points with coordinates
(t,r) and (t+dt, r+dr). Different observers measure different curvatures.
We'll be talking of intrinsic curvatures. To relate the results there's a
set of transformations called Diffeomorphism Symmetry relating one view
point to another. Spacetime is also elastic but rather stiff.
One of the features of gtr which many people initially find strange and
even disturbing is that "spacetime" is fixed once and for all; it
describes the entire history of motion of each and every small body or
particle of matter or mass-energy in the universe, leaving no room for
"free will", such as the decision of our astronauts to fire their
retrorockets at some time. (This would cause them to feel an acceleration
for the period of time during which the rockets are firing, and the world
line of the capsule would not be a geodesic during this period, but would
"bend" in spacetime.)
WHAT IS S-T?
In the context of the general theory of relativity, spacetime itself is
normally regarded as a field, i.e., an extended physical entity, which not
only acts upon material objects but is also acted upon by them, so the
absolute-relational distinction (in physical theories) has no clear
meaning. However, it remains possible to regard fields as only
representations of effects, and to insist on materiality for ontological
objects, in which case the absolute-relational question remains both
relevant and unresolved.
One half of general relativity is the statement that any object moves
along paths of maximum proper time, i.e., along geodesics. The other
half is contained in the expression: the sum of all three 'proper'
sectional 'spatial' curvatures at a point is given by
K_(12) + K_(23) + K_(31) = 8piG/c^4 W^(0).
where W^(0) is the 'proper' energy density at the point, and this
statement is valid for every observer. The lower indices indicate the
mixed curvatures defined by the three orthogonal directions 1, 2 and 3.
This is all of general relativity in one paragraph. The single equation
above must be expanded to ten equations, Einstein's field equations, then
things get mathematically more involved. Einstein's eqns are the central
law of physics in gtr. Einstein's equation says that the deviation of the
actual volume from an euclidean value is given by the "moment of inertia"
of the amount of mass-energy inside our "nearly spherical" surface of
"radius" r!
CURVATURE
There are many ways to understand curvature. One is that in curved
spacetimes, initially parallel geodesics (curves in which the distance
between two nearby events is a local minimum, taken over "nearby"
paths) tend to diverge (negative curvature) or converge (positive
curvature). Another is that if you transport a vector in a loop in
spacetime (this is not a physical motion, since nothing physical ever
moves in spacetime, but mathematically it makes perfect sense), keeping it
parallel to itself, it usually comes back rotated slightly. Still another
way of thinking about curvature involves the fact that a certain kind of
geometrically meaningful "second derivative" depends on the order of
differentiation, and the difference between the two results is,
essentially, the curvature. In two dimensional manifolds, curvature is a
scalar field; in higher dimensional manifolds, it is a tensor field (at
each point, such a field defines a multilinear map, analogous to a linear
transformation but with more than one vector "inputs").
SYMMETRY
Each point on the surface of an ordinary sphere is perfectly symmetrical
with every other point, but there is no difficulty imagining the arbitrary
(random) selection of a single point on the surface, because we can define
a uniform probability density on this surface. However, if we begin with
an infinite flat plane (like in Minkowski space), where again each point
is perfectly symmetrical with every other point, we face an inherent
difficulty, because there does not exist a perfectly uniform probability
density distribution over an infinite surface. Hence, if we select one
particular point on this infinite flat plane, we can't claim, even in
principle, to have chosen from a perfectly uniform distribution.
Therefore, the original empty infinite flat plane was not perfectly
symmetrical after all, at least not with respect to our selection of
individual points. This shows that the very idea of selecting a point from
a pre-existing perfectly symmetrical infinite manifold is, in a sense,
self-contradictory.
The asymmetry due to the presence of an infinitesimal inertial particle in
flat Minkowski spacetime is purely circumstantial, because the spacetime
is considered to be unaffected by the presence of this particle. However,
according to general relativity, the presence of any inertial entity
disturbs the symmetry of the manifold even more profoundly, because it
implies an intrinsic curvature of the spacetime manifold, i.e., the
manifold takes on an intrinsic shape that distinguishes the location and
rest frame of the particle. For a single non-rotating uncharged particle
the resulting shape is Schwarzschild spacetime, which obviously exhibits a
distinguished center and rest frame (the frame of the central mass).
Indeed, this spacetime exhibits a preferred system of coordinates, namely
those for which the metric coefficients are independent of the time
coordinate.
THE FIELD VARIABLES
Since the field variables of general relativity are the metric
coefficients themselves, we are naturally encouraged to think that there
is no a priori distinguished system of reference in the physical spacetime
described by general relativity, and that it is only the contingent
circumstance of a particular distribution of inertial entities that may
distinguish any particular frame or state of motion. In other words, it's
tempting to think that the spacetime manifold is determined solely by its
"contents", i.e., that the left side of G_uv = 8piT_uv (constants G and c
in this equation were made equal to 1) is determined by the right side.
However, this is not actually the case (as Einstein and others realized
early on), and to understand why, it's useful to review what is involved
in actually solving the field equations of general relativity as an
initial-value problem. Mostly important is the fact that the Einstein
tensor G_uv is symmetric (meaning it has ten independent components) and
its divergence vanishes (describes a conserved quantity): This property
allowed Einstein to relate it to mass and energy in mathematical language.
WHAT IS GR?
General relativity is not a relational theory of motion in the sense that
general relativity does not correlate all physical effects with the
relations between material bodies, but rather with the relations between
objects (including fields) and the absolute background metric, which is
affected by, but is not determined by, the distribution of objects (except
arguably in closed cosmological models). Thus relativity, no less than
Newtonian mechanics, relies on spacetime as an absolute entity in itself,
exerting influence on fields and material bodies.
In this way relativity very quickly disappointed its early
logical-positivist supporters when it became clear that it was not, and
never had been, a relational theory of motion, ala Leibniz, Berkeley, or
Mach. Initially even Einstein was "scandalized" by the Schwarzschild and
de Sitter solutions, which represent complete metrical manifolds with only
one material object or none at all (respectively). These examples showed
that spacetime in the theory of relativity cannot simply be regarded as
the totality of the extrinsic relations between material objects (and
non-gravitational fields), but is a primary _physical entity_ of the
theory, with its own absolute properties, most notably the metric with its
related invariants, at each point. Indeed this was Einstein's eventual
answer to Mach's critique of pre-relativity physics. Mach had complained
that it was unacceptable for our theories to contain elements (such as
spacetime) that act on (i.e., have an effect on) other things, but that
are not acted upon by other things. Mach, and the other relationalists
before him, naturally expected this to be resolved by eliminating
spacetime, i.e., by denying that an entity called "spacetime" acts in any
physical way. To Mach's surprise (and unhappiness), the theory of
relativity actually did just the opposite - it satisfied Mach's criticism
by instead making spacetime a full-fledged element of the theory, acted
upon by other objects. By so doing, Einstein believed he had responded to
Mach's critique, but of course Mach hated it, and said so.
Early in his career, Einstein was sympathetic to the idea of relationism,
and entertained hopes of banishing absolute space from physics but, like
Newton before him, he was forced to abandon this hope in order to produce
a theory that satisfactorily represents our observations.
THE EQUATIONS
The ten algebraically independent field equations represented by G_uv =
8piT_uv involve the values of the ten independent metric coefficients and
their first and second derivatives with respect to four spacetime
coordinates. If we're given the values of the metric coefficients
throughout a 3D spacelike "slice" of spacetime at some particular value of
the time coordinate, we can directly evaluate the first and second
derivatives of these components with respect to the space coordinates in
this "slice". This leaves only the first and second derivatives of the ten
metric with respect to the time coordinate as unknown quantities in the
ten field equations. It might seem that we could arbitrarily specify the
first derivatives, and then solve the field equations for the second
derivatives, enabling us to "integrate" forward in time to the next
timeslice, and then repeat this process to predict the subsequent
evolution of the metric field. However, the structure of the field
equations does not permit this, because four of the ten field equations
(namely, G_0v = 8piT_0v with v = 0,1,2,3) contain only the first
derivatives with respect to the time coordinate x_0, so we can't
arbitrarily specify the g_uv and their first derivatives with respect to
x_0 on a surface of constant x_0. These ten first derivatives, alone, must
satisfy the four G_0v conditions on any such surface, so before we can
even pose the initial value problem, we must first solve this subset of
the field equations for a viable set of initial values. Although these
four conditions constrain the initial values, they obviously don't fully
determine them, even for a given distribution of T_uv.
THE COMPLICATIONS
Once we've specified values of the g_uv and their first derivatives with
respect to x_0 on some surface of constant x_0 in such a way that the four
conditions for G_0v are satisfied, the four contracted Bianchi identities
ensure that these conditions remain satisfied outside the initial surface,
provided only that the remaining six equations are satisfied everywhere.
However, this leaves only six independent equations to govern the
evolution of the ten field variables in the x_0 direction. As a result,
the second derivatives of the g_uv with respect to x_0 appear to be
underdetermined. In other words, given suitable initial conditions, we're
left with a four-fold ambiguity. We must arbitrarily impose four more
conditions on the system in order to uniquely determine a solution. This
was to be expected, because the metric coefficients depend not only on the
absolute shape of the manifold, but also on our choice of coordinate
systems, which represents four degrees of freedom. Thus, the field
equations actually determine an equivalence class of solutions,
corresponding to all the ways in which a given absolute metrical manifold
can be expressed in various coordinate systems. In order to actually
generate a solution of the initial value problem, we need to impose four
"coordinate conditions" along with the six "dynamical" field equations.
The conditions arise from any proposed system of coordinates by expressing
the metric coefficients g_0v in terms of these coordinates (which can
always be done for any postulated system of coordinates), and then
differentiating these four coefficients twice with respect to x_0 to give
four equations in the second derivatives of these coefficients.
A. Einstein used to say that gr only provides the understanding of one
side of the field equations (G_uv = 8piT_uv ), but not of the other. Can
Tim Chow see now which side AE meant?
THE MANIFOLD
Notwithstanding the four-fold ambiguity of the dynamical field equations,
which is just a descriptive rather than a substantive ambiguity, it's
clear that the manifold is a definite absolute entity, and its overall
characteristics and evolution are determined not only by the postulated
T_uv and the field equations, but also by the conditions specified on the
initial timeslice. As noted above, these conditions are constrained by the
field equations, but are by no means fully determined. We are still
required to impose largely arbitrary conditions in order to fix the
absolute background spacetime. This state of affairs was disappointing to
Einstein, because he recognized that the selection of a set of initial
conditions is tantamount to stipulating a preferred class of reference
systems, precisely as in Newtonian theory, which is "contrary to the
spirit of the relativity principle" (referring presumably to the
relational ideas of Mach).
SOLUTIONS
There are multiple distinct vacuum solutions of the field equations, some
with gravitational waves and even geons (temporarily) zipping around, and
some not. Even more ambiguity arises when we introduce mass, as Godel
showed with his cosmological solutions in which the average mass of the
universe is rotating with respect to the spacetime background. These
examples just highlight the fact that general relativity can no more
dispense with the arbitrary stipulation of a preferred class of reference
systems (the inertial systems) than could Newtonian mechanics or special
relativity.
This is clearly illustrated by Schwarzschild spacetime, which (according
to Birkhoff's theorem) is the essentially unique spherically symmetrical
solution of the field equations. Clearly this cosmological model, based on
a single spherically symmetrical mass in an otherwise empty universe, is
"contrary to the spirit of the relativity principle" because, as noted
earlier, there is an essentially unique time coordinate for which the
metric coefficients are independent of time.
ISOMETRIES
Translation along a vector that leaves the metric formally unchanged is
called an isometry, and a complete vector field of isometries is called a
Killing vector field. Thus the Schwarzschild time coordinate t constitutes
a Killing vector field over the entire manifold, making it a highly
distinguished time coordinate, no less than Newton's absolute time. In
both special relativity and Newtonian physics there is an infinite class
of operationally equivalent systems of reference at any point, but in
Schwarzschild spacetime there is an essentially unique global coordinate
system with respect to which the metric coefficients are independent of
time, and this system is related in a definite way to the inertial class
of reference systems at each point. Thus, in the context of this
particular spacetime, we actually have a much stronger case for a
meaningful notion of absolute rest than we do in Newtonian spacetime or
special relativity, both of which rest naively on the principle of
inertia, and neither of which acknowledges the possibility of variations
in the properties of spacetime from place to place (let alone under
velocity transformations).
THE END
Incidently, Einstein was initially hostile to Minkowski's geometric
viewpoint (possibly because Minkowski, exasperated by his talented young
student's refusal to turn in any homework, once told him "Einstein, you
are a smart boy, a very smart boy, but you are a lazy dog!"). Einstein
complained wryly to his friends that "since the mathematicians have taken
up my theory, I myself no longer understand it!" But he soon came to
value the geometric viewpoint and, typically, not only embraced
Minkowski's ideas but went far beyond them in the creation of his general
theory of relativity (gtr), which incorporated both str and a
revolutionary new theory of gravitation, in which Minkowski geometry is
the geometry of "tangent" planes to a curved spacetime; in which
gravitation is represented by the curvature itself.
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